We are tasked with showing that the function $[·]_\alpha : V \to \mathbb{R}^n$, defined by $[x]_\alpha = (a_1, a_2, \dots, a_n)$, where $x = a_1 x_1 + a_2 x_2 + \dots + a_n x_n$, is a linear transformation.

To show that $[·]_\alpha$ is a linear transformation, we need to verify two properties:

1. Additivity: $[x + y]_\alpha = [x]_\alpha + [y]_\alpha$
2. Homogeneity: $[cx]_\alpha = c[x]_\alpha$, for any scalar $c \in \mathbb{R}$

Step 1: Additivity

Let $x, y \in V$ be any two vectors. Since $x = a_1 x_1 + a_2 x_2 + \dots + a_n x_n$ and $y = b_1 x_1 + b_2 x_2 + \dots + b_n x_n$ for some scalars $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$, we need to show:

[ [x + y]\alpha = [x]\alpha + [y]_\alpha ]

Consider $x + y$:

$x + y = (a_1 x_1 + a_2 x_2 + \dots + a_n x_n) + (b_1 x_1 + b_2 x_2 + \dots + b_n x_n)$

By the linearity of vector addition, we can group the terms:

$x + y = (a_1 + b_1) x_1 + (a_2 + b_2) x_2 + \dots + (a_n + b_n) x_n$

Thus, the coordinates of $x + y$ with respect to the basis $\alpha$ are:

$[x + y]_\alpha = ((a_1 + b_1), (a_2 + b_2), \dots, (a_n + b_n))$

Now, the coordinates of $x$ and $y$ are:

[ [x]\alpha = (a_1, a_2, \dots, a_n), \quad [y]\alpha = (b_1, b_2, \dots, b_n) ]

Adding these two coordinate vectors component-wise:

[ [x]\alpha + [y]\alpha = (a_1 + b_1, a_2 + b_2, \dots, a_n + b_n) ]

Thus, we have:

[ [x + y]\alpha = [x]\alpha + [y]_\alpha ]

This shows that $[·]_\alpha$ satisfies additivity.

Step 2: Homogeneity

Let $x \in V$ be a vector and $c \in \mathbb{R}$ a scalar. We need to show:

[ [cx]\alpha = c[x]\alpha ]

Recall that $x = a_1 x_1 + a_2 x_2 + \dots + a_n x_n$. Then,

$cx = c(a_1 x_1 + a_2 x_2 + \dots + a_n x_n) = (c a_1) x_1 + (c a_2) x_2 + \dots + (c a_n) x_n$

Thus, the coordinates of $cx$ with respect to the basis $\alpha$ are:

$[cx]_\alpha = (c a_1, c a_2, \dots, c a_n)$

On the other hand, the coordinates of $x$ with respect to $\alpha$ are:

$[x]_\alpha = (a_1, a_2, \dots, a_n)$

Multiplying the vector $[x]_\alpha$ by $c$:

$c[x]_\alpha = (c a_1, c a_2, \dots, c a_n)$

Thus, we have:

[ [cx]\alpha = c[x]\alpha ]

This shows that $[·]_\alpha$ satisfies homogeneity.

Conclusion

Since we have shown both additivity and homogeneity, we can conclude that $[·]_\alpha : V \to \mathbb{R}^n$ is a linear transformation.

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Would you like a further explanation of any step, or is there anything else you'd like to explore?

Here are some related questions:

1. How do you show that a different function on vector spaces is a linear transformation?
2. What is the relationship between a basis of a vector space and coordinate transformations?
3. How can we represent vector space homomorphisms using coordinate systems?
4. What are the implications of the linearity of coordinate functions in higher dimensions?
5. How do we use the concept of linear transformations in applications like computer graphics?

Tip: When proving linearity, always verify the two key properties (additivity and homogeneity) for your function!